The optimal power flow (OPF) is a subject of interest in power system operations, scheduling, and planning. The main objective of an OPF solver is to determine the optimal steady-state operation of an electric power system while satisfying technical and economic constraints. With the structural deregulation of electric power systems, OPF is becoming a basic tool in the power market. Existing solvers and programs for OPF computation are mostly focused on achieving an optimal solution to the study objective for the pre-contingency system (base case system); i.e., the electric power system without any security/stability considerations. However, it is crucial for the OPF solutions being not only economic and but also secure. The terms “secure” and “security” as used herein are interchangeable with the terms “stable” and “stability,” respectively. More specifically, a power system is secure if it is able to maintain a normal and stable operation when encountering contingencies, where contingencies are discrete events such as failure of devices (e.g., lines, generators, shunts, etc.). A widely accepted system security concept is the so-called “N-1 security;” that is, only one device of the power network fails at a time for a single credible contingency. A contingency is “credible” if its occurrence is plausible and/or falls in a range of likelihood. The “N-1 security” contingency standard has been established by the North American Electric Reliability Corporation (NERC) and is required to be complied by power utilities. Accordingly, an electric power system is required to be operated in a way such that it can survive the occurrence of any single credible contingency where any single component in the system goes offline suddenly.
Mathematically, an OPF problem is modeled as a nonlinear programming (NLP) problem, which usually minimizes the total generation dispatch cost, transmission loss, or their combination subject to a set of equality and inequality constraints. From a computational viewpoint, an OPF problem is a large-scale non-convex NLP problem, in which both the objective function and constraint functions can be nonlinear. An OPF problem becomes a mixed-integer NLP when discrete control variables such as transformer taps, shunt capacitor banks and Flexible AC transmission system (FACTS) devices are taken into account. Furthermore, if transient stability constraints are considered, then an OPF problem can be expressed as a set of large-scale differential-algebraic equations (DAE).
An OPF problem can be formulated as a general constrained nonlinear optimization problem of the following form:minx=(xs,xc)f(xs,xc)s·th(x)=0g(x)≤0xl≤x≤xu  (1)where, xs=[θ1, θ2, . . . , θn, V1, V2, . . . , Vn]T is the state vector composed of variables of system running states including bus voltage amplitudes and phase angles;xc=[Pg1, Pg2, . . . , Pgm, Qg1, Qg2, . . . , Qgm, tap1, . . . ] is the control vector which is composed of control variables such as generator real and reactive power outputs, transformer taps, shunt capacitor banks, FACTS control variables, and so on; g(x) represents nonlinear equality constraints required for power flow balance at each bus; h(x) represents nonlinear inequality constraints composed of functional and operational constraints such as power flow limits over transmission lines and transformers, limits on VAR (voltage-ampere reactive, which is a unit used to measure reactive power in an AC electric power system) injections for reactive control buses and real power injections for the slack buses; xl and xu are the lower and upper bounds to be imposed on state and control variables.
There has been a wealth of research efforts focused on developing effective and robust nonlinear programming (NLP) methods for solving general nonlinear optimization problems of the form (1) and applications of these methods to solving OPF problems with a full set of nonlinear equality and inequality constraints. Existing methods for solving OPF problems either through solving an approximated linear program (LP) or through solving the NLP directly. LP based solvers are generally fast and very robust. However, since the LP is only an approximate to the true OPF problem, the resulting LP solutions can be an inaccurate approximate to the true OPF solution. Moreover, some information, such as the locationally marginal prices (LMP) that is required by power markets cannot be accurately obtained. NLP based solvers, on the other hand, can result in accurate OPF solutions and LMP values, at the expense of increased solver complexity and decreased solver robustness. More specifically, due to the nonlinearity and nonconvexity of the objective and constraint functions of the OPF problem, the convergence of the solver is usually not guaranteed, even though an OPF solution indeed exists. When security criteria are satisfied for the OPF solution, the resulting optimization problem becomes security-constrained optimal power flow (SCOPF). Therefore, the goal of an SCOPF problem is to ensure that the system operate properly under both the pre-contingency (base case system) and post-contingency conditions. Existing SCOPF methods usually try to solve an augmented OPF problem to co-optimize among the involved contingences. Complexity, in terms of the number of decision variables and the number of equality and inequality constraints, of the resulting OPF problem grows rapidly as the number of contingencies under study increases. However, such increased complexity could result in numerical issues including poor convergence, rapidly increased consumption of CPU time and other computational resources. For example, in order to perform a security-constrained OPF analysis on a 10,000-bus system with 100 contingencies, the augmented OPF problem to be solved will have millions of variables and nonlinear constraints. Therefore, the augmented OPF problem can easily become intractable or even impossible by the available computational resources as the scale of the system or the number of contingencies increases.